Metropolitan and peripheral theories attempt to explain why imperialism happened the way that it did in various places. The difference between the two lies in what kinds of factors each thinks were responsible for how imperialism played out.
Metropolitan theories argue that imperialism was caused by factors that have to do with the imperial country. The country that conquers others is called the "metropole," which is why these are called metropolitan theories. John Hobson was a metropolitan theorist in that he argued that it was British manufacturing needs that led to imperialism.
Peripheral theories argue that the conditions in the conquered countries (the are called the "periphery") were important. They note, for example, that some possessions were ruled directly by the metropole while others were ruled indirectly through local elites. They say that the difference was caused by local conditions. In places that had stable local governments that were willing to cooperate with the metropole, the imperial country did not have to rule directly. However, if there were no such conditions, it would have to rule its possessions directly in a "formal empire." Gallagher and Robinson are examples of scholars who hold to this idea.